LOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
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1 LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 Introducton In dummy regresson varable models, t s assumed mplctly that the dependent varable Y s quanttatve whereas the explanatory varables are ether quanttatve or qualtatve. There are certan type of regresson models n whch the dependent or response varable s dchotomous n nature, takng a or 0 value. Suppose one wants to study the labor-force partcpaton of adult males as a functon of the unemployment rate, average wage rate, famly ncome, educaton etc. A person s ether n the labor force or not. Hence, the dependent varable, labor-force partcpaton, can take only two values: f the person s n the labor force and 0 f he or she s not. There are several examples where the dependent varable s dchotomous. Suppose on wants to study the unon membershp status of the college professors as a functon of several quanttatve and qualtatve varables. A college professor ether belongs to a unon or does not. Therefore, the dependent varable, unon membershp status, s a dummy varable takng on values 0 or, 0 meanng no unon membershp and meanng unon membershp. Smlarly, other examples can be ownershp of a house: a famly owns a house or t does not, t has dsablty nsurance or t does not, a certan drug s effectve n curng an llness or t s not, decson of a frm to declare a dvdend or not, Presdent decdes to veto a bll or accept t, etc. A unque feature of all these examples s that the dependent varable s of the type whch elcts a yes or no response. There are specal estmaton / nference problems assocated wth such models. The most commonly used approaches to estmatng such models are the Lnear Probablty model, the Logt model and the Probt model. There are certan problems assocated wth the estmaton of Lnear Probablty Models such as: ( Non-normalty of the Dsturbances (U s ( Heteroscedastc varances of the dsturbances ( Nonfulfllment of 0 E(Y X ( Possblty of Yˆ lyng ousde the 0 - range (v Questonable value of R 2 as a measure of goodness of ft; and Lnear Probablty Model s not logcally a very attractve model because t assumes that P =E(Y= X ncreases lnearly wth X, that s, the margnal or ncremental effect of X remans constant throughout. Ths seems sometmes very unrealstc. Therefore, there s a need of a probablty model that has two features: ( as X ncreases, P = E(Y = X ncreases but never steps outsde the 0- nterval, and (2 the relatonshp between P and X s non-lnear, that s, approaches one whch approaches zero at slower and slower rates as X gets small and approaches one at slower and slower rates as X gets very large.
2 Logt Analyss 2. Logt Model Logt regresson (logt analyss s a un/multvarate technque whch allows for estmatng the probablty that an event occurs or not, by predctng a bnary dependent outcome from a set of ndependent varables. In an example of home ownershp where the dependent varable owns a house or not n relaton to ncome, the lnear probablty Model depcted t as P = E(Y = X = β + β 2 X Where X s the ncome and Y = means that the famly owns a house. Let us consder the followng representaton of home ownershp: P = E(Y = X = exp[ ( β +β 2 + X ] exp( Z...(2. where Z = β + β 2 X Ths equaton ( s known as the (cumulatve logstc dstrbuton functon. Here Z ranges from - to + ; P ranges between 0 and ; P s non-lnearly related to Z (.e. X thus satsfyng the two condtons requred for a probablty model. In satsfyng these requrements, an estmaton problem s created because P s nonlnear not only n X but also n the β s. Ths means that one cannot use OLS procedure to estmate the parameters. Here, P s the probablty of ownng a house and s gven by exp( Z Then (- P, the probablty of not ownng a house, s (- P = exp( Z Therefore, one can wrte P exp( Z = P exp( Z (2.2 P /(- P s the odds rato n favour of ownng a house.e; the rato of the probablty that a famly wll own a house to the probablty that t wll not own a house. Takng natural log of (2, we obtan L = ln [P /( - P ] = Z = β + β 2 X (2.3 2
3 Logt Analyss That s, the log of the odds rato s not only lnear n X, but also lnear n the parameters. L s called the Logt. 2. Features of Logt Model ( As P goes from 0 to, the logt L goes from - to +. That s, although the probabltes le between 0 and, the logts are not so bounded. ( Although L s lnear n X, the probabltes themselves are not. ( The nterpretaton of the logt model s as follows: β 2, the slope, measures the change n L for a unt change n X,.e. t tells how the log odds n favour of ownng a house change as ncome changes by a unt. The ntercept β s the value of the log odds n favour of ownng a house f ncome s zero. (v Gven a certan level of ncome, say X *, f we actually want to estmate not the odds n favour of ownng a house but the probablty of ownng a house tself, ths can be done drectly once the estmates of β and β 2 are avalable. (v The lnear probablty model assumes that P s lnearly related to X, the logt model assumes that the log of odds rato s lnearly related to X 2.2 Estmaton of Logt Model In order to estmate the logt model, we need apart from X, the values of logt L. By havng data at mcro or ndvdual level, one cannot estmate (3 by OLS technque. In such stuatons, one has to resort to Maxmum Lkelhood method of estmaton. In case of grouped data, correspondng to each ncome level X, there are N famles among whch n are possessng a house. Therefore, one needs to compute n Pˆ = N Ths relatve frequency s an estmate of true P correspondng to each X. Usng the estmated P, one can obtan the estmated logt as Lˆ = ln [ P /( P ] L = Z = β + β 2 X 2.2. Steps n Estmatng Logt Regresson ( Compute the estmated probablty of ownng a house for each ncome level X, as n Pˆ = N ( For each X, obtan the logt as ( Lˆ = ln [Pˆ /( Pˆ ] (v Transform the logt regresson n order to resolve the problem of heteroscedastcty as follows: W L = β W + β 2 W X + W U (2.4 where the weghts W = N Pˆ /( Pˆ. 3
4 Logt Analyss (v Estmate (2.4 by OLS (WLS s OLS on transformed data (v Establsh confdence ntervals and / or test hypothess n the usual OLS framework. All the conclusons wll be vald strctly only when the sample s reasonably large. 2.3 Merts of Logt Model ( Logt analyss produces statstcally sound results. By allowng for the transformaton of a dchotomous dependent varable to a contnuous varable rangng from - to +, the problem of out of range estmates s avoded. ( The logt analyss provdes results whch can be easly nterpreted and the method s smple to analyse. ( It gves parameter estmates whch are asymptotcally consstent, effcent and normal, so that the analogue of the regresson t-test can be appled. 2.4 Demerts ( As n the case of Lnear Probablty Model, the dsturbance term n logt model s heteroscedastc and therefore, we should go for Weghted Least Squares. ( N has to be farly large for all X and hence n small sample; the estmated results should be nterpreted carefully. ( As n nay other regresson, there may be problem of multcollnearty f the explanatory varables are related among themselves. (v As n Lnear Probablty Models, the conventonally measured R 2 s of lmted value to judge the goodness of ft. 2.5 Applcaton of Logt Model Analyss ( It can be used to dentfy the factors that affect the adopton of a partcular technology say, use of new varetes, fertlzers, pestcdes etc, on a farm. ( In the feld of marketng, t can be used to test the brand preference and brand loyalty for any product. ( Gender studes can use logt analyss to fnd out the factors whch affect the decson makng status of men/women n a famly. 3. Probt Model In order to explan the behavour of a dchotomous dependent varable we have to use a sutably chosen Cumulatve Dstrbuton Functon (CDF. The logt model uses the cumulatve logstc functon. But ths s not the only CDF that one can use. In some applcatons, the normal CDF has been found useful. The estmatng model that emerges from the normal CDF s known as the Probt Model or Normt Model. Let us assume that n home ownershp example, the decson of the th famly to own a house or not depends on unobservable utlty ndex I, that s determned by the explanatory varables n such a way that the larger the value of ndex I, the greater the probablty of the famly ownng a house. The ndex I can be expressed as I = β + β 2 X, where X s the ncome of the th famly. (3. 4
5 Logt Analyss 3. Assumpton of Probt Model For each famly there s a crtcal or threshold level of the ndex (I *, such that f I exceeds I *, the famly wll own a house otherwse not. But the threshold level I * s also not observable. If t s assumed that t s normally dstrbuted wth the same mean and varance, t s possble to estmate the parameters of (3. and thus get some nformaton about the unobservable ndex tself. In Probt Analyss, the unobservable utlty ndex (I s known as normal equvalent devate (n.e.d or smply Normt. Snce n.e.d. or I wll be negatve whenever P < 0.5, n practce the number 5 s added to the n.e.d. and the result so obtaned s called the Probt.e; Probt = n.e.d + 5 = I + 5 In order to estmate β and β 2, (3. can be wrtten as I = β + β 2 X + U ( Steps nvolved n Estmaton of Probt Model ( Estmate P from grouped data as n the case of Logt Model,.e., n Pˆ = N ( Usng P, obtan n.e.d (I from the standard normal CDF,.e. I = β + β 2 X, ( Add 5 to the estmated I to convert them nto probts and use the probts thus obtaned as the dependent varable n (3.2. (v As n the case of Lnear Probablty Model and Logt Model, the dsturbance term s heteroscedastc n Probt Model also. In order to get effcent estmates, one has to transform the model. (v After transformaton, estmate (3.2 by OLS. 4. Logt versus Probt ( The chef dfference between logt and probt s that logstc has slghtly flatter tals.e; the normal or probt curve approaches the axes more quckly than the logstc curve. ( Qualtatvely, Logt and Probt Models gve smlar results, the estmates of parameters of the two models are not drectly comparable. 5
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